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Then after a brief exposition of the contact geometry, we explore its connection with the non-standard Hamiltonian structures. We present the formulation of the Levinson\u2013Smith equation in terms of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) method and also study the gradient-type flow. We give a geometric formulation of GENERIC and apply this to general Levinson\u2013Smith equations.<\/jats:p>","DOI":"10.1007\/s00332-024-10090-y","type":"journal-article","created":{"date-parts":[[2024,9,25]],"date-time":"2024-09-25T20:46:20Z","timestamp":1727297180000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Levinson\u2013Smith Dissipative Equations and Geometry of GENERIC Formalism and Contact Hamiltonian Mechanics"],"prefix":"10.1007","volume":"34","author":[{"given":"Jos\u00e9 F.","family":"Cari\u00f1ena","sequence":"first","affiliation":[]},{"given":"A. 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